Bisectors, Medians, Altitudes

1
Bisectors, Medians, Altitudes

• Neither you nor the world knows what you can do
  until you have tried. -- Ralph Waldo Emerson

• Chapter 5 Section 1
• Learning Goal Understand and Draw the 4 special
  segments of a Triangle

2
Bisectors

• There are 2 different types of bisectors
• (draw the indicated bisector of each figure)

Segment Bisector A line, segment or ray that cuts
a segment in half Perpendicular Bisector
Angle Bisector A line, segment or ray that cuts
an angle in half
3
Perpendicular Bisector
Given Bisector Show distance from a point on
the bisector to the endpoints
4
Perpendicular Bisector

• If AB is the ? bisector of XY, ZY 8, and ?ZXY
  is a right triangle, find XA
• If XZ intersects GB, and ZB 3, what is GZ?

X
A
6
Y
G
Z
B
Not Enough Information !!!
5
Angle Bisector
L
Which Set of Lines demonstrate this Theorem?
K
P
M
6
Angle Bisector
L
How Bout Now?
K
Remember the definition of Distance from a point
to a line.
P
M
7
Angle Bisectors

• Find the value of each variable, given the angle
  bisector.

a v 80
y
z
110
x
79
x 70 y 31 z 39
12
12
8
a
8
Medians

• A Median is a segment whose endpoints are a
  vertex and the midpoint of the opposite side of a
  triangle.

x
y
Draw the medians of the triangles below
a
b
y
x
Does x y ???
a
b
Does a b ???
9
Altitude

• The Altitude of a triangle is a segment from a
  vertex to the line containing the opposite side,
  and perpendicular to that side.

Draw the Altitudes for the Triangle
Remember How to find the area of a Triangle
P
B
A ½ b h
A
10
Special Segments

• Identify the Special Segment in the picture
• 1. 2.
• 3. 4.

11
Find Every measurement you can
Given an altitude Given the ? bisector
Given an Angle Bisector Given a Median
51
10
10
27
46
31
8
120
20
12
Proof
Definition of Angle Bisector
13
Homework

• Pages 275 277 4 8 (even), 11, 13, 22, 26,
  36, and 37
• (9 problems)